A manufacturer produces two types of products, 1 and 2, at production levels of $x_1$ and $x_2$ respectively. The profit is $2x_1+5x_2$. Constraints are $x_1+3x_2\le 40$, $x_1+x_2\le 10$, $x_1\ge 0,x_2\ge 0$. The maximum profit is:

Objective function: $Z=2x_1+5x_2$.Corner points of the feasible region defined by constraints:Intersection of $x_1+3x_2=40$ and $x_1+x_2=10$:Subtracting equations: $2x_2=30⟹x_2=15$. Then $x_1=-5$.Since $x_1\ge 0$, this point is outside the feasible region.Boundary intercepts:For $x_1+3x_2\le 40$: $(40,0)$ and $(0,40/3\approx 13.33)$.For $x_1+x_2\le 10$: $(10,0)$ and $(0,10)$.The intersection of the feasible region involves the most restrictive constraints:Corner points are $(0,0),(10,0),(0,10)$.$Z(0,0)=0$$Z(10,0)=2(10)+0=20$$Z(0,10)=0+5(10)=50$.Maximum profit is $50$ at $(0,10)$.